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Clique-Based Deletion-Correcting Codes via Penalty-Guided Clique Search

Aniruddh Pandav, Rajshekhar V Bhat

TL;DR

This work tackles constructing $d$-deletion-correcting binary codes by casting the problem as a Maximum Clique Problem on a deletion-compatibility graph defined via the LCS distance. A Penalty-Guided Clique Search (PGCS) heuristic is proposed to efficiently find large cliques, yielding larger codebooks than prior graph-based heuristics for $n$ up to 14 and $d$ up to 3, with some instances matching known optimal sizes. An optimized LCS-based decoder for segmented reception is introduced, employing symbol-count filtering and early termination to reduce decoding complexity while preserving exact recovery. The combination of PGCS-based construction and efficient decoding offers practical deletion-correcting codes with improved performance, though graph construction remains the main scalability bottleneck. The results highlight a path toward scalable, high-performance deletion-correcting codes for moderate block lengths.

Abstract

We study the construction of $d$-deletion-correcting binary codes by formulating the problem as a Maximum Clique Problem (MCP). In this formulation, vertices represent candidate codewords and edges connect pairs whose longest common subsequence (LCS) distance guarantees correction of up to $d$ deletions. A valid codebook corresponds to a clique in the resulting graph, and finding the largest codebook is equivalent to identifying a maximum clique. While MCP-based formulations for deletion-correcting codes have previously been explored, we demonstrate that applying Penalty-Guided Clique Search (PGCS), a lightweight stochastic clique-search heuristic inspired by Dynamic Local Search (DLS), consistently yields larger codebooks than existing graph-based heuristics, including minimum-degree and coloring methods, for block lengths $n = 8,9,\dots,14$ and deletion parameters $d = 1,2,3$. In several finite-length regimes, the resulting codebooks match known optimal sizes and outperform classical constructions such as Helberg codes. For decoding under segmented reception, where codeword boundaries are known, we propose an optimized LCS-based decoder that exploits symbol-count filtering and early termination to substantially reduce the number of LCS evaluations while preserving exact decoding guarantees. These optimizations lead to significantly lower average-case decoding complexity than the baseline $O(|C| n^2)$ approach.

Clique-Based Deletion-Correcting Codes via Penalty-Guided Clique Search

TL;DR

This work tackles constructing -deletion-correcting binary codes by casting the problem as a Maximum Clique Problem on a deletion-compatibility graph defined via the LCS distance. A Penalty-Guided Clique Search (PGCS) heuristic is proposed to efficiently find large cliques, yielding larger codebooks than prior graph-based heuristics for up to 14 and up to 3, with some instances matching known optimal sizes. An optimized LCS-based decoder for segmented reception is introduced, employing symbol-count filtering and early termination to reduce decoding complexity while preserving exact recovery. The combination of PGCS-based construction and efficient decoding offers practical deletion-correcting codes with improved performance, though graph construction remains the main scalability bottleneck. The results highlight a path toward scalable, high-performance deletion-correcting codes for moderate block lengths.

Abstract

We study the construction of -deletion-correcting binary codes by formulating the problem as a Maximum Clique Problem (MCP). In this formulation, vertices represent candidate codewords and edges connect pairs whose longest common subsequence (LCS) distance guarantees correction of up to deletions. A valid codebook corresponds to a clique in the resulting graph, and finding the largest codebook is equivalent to identifying a maximum clique. While MCP-based formulations for deletion-correcting codes have previously been explored, we demonstrate that applying Penalty-Guided Clique Search (PGCS), a lightweight stochastic clique-search heuristic inspired by Dynamic Local Search (DLS), consistently yields larger codebooks than existing graph-based heuristics, including minimum-degree and coloring methods, for block lengths and deletion parameters . In several finite-length regimes, the resulting codebooks match known optimal sizes and outperform classical constructions such as Helberg codes. For decoding under segmented reception, where codeword boundaries are known, we propose an optimized LCS-based decoder that exploits symbol-count filtering and early termination to substantially reduce the number of LCS evaluations while preserving exact decoding guarantees. These optimizations lead to significantly lower average-case decoding complexity than the baseline approach.
Paper Structure (26 sections, 1 theorem, 22 equations, 2 tables, 4 algorithms)

This paper contains 26 sections, 1 theorem, 22 equations, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Solution
  4. Preliminaries
  5. Longest Common Subsequence
  6. Cliques and the Maximum Clique Problem
  7. LCS-Based Characterization of Deletion Correction
  8. Construction
  9. Reduction to Maximum Clique Problem
  10. LCS Computation
  11. Graph Construction and Clique Finding
  12. Penalty-Guided Clique Search
  13. Decoding
  14. LCS-Based Decoder
  15. Decoder Correctness
  16. ...and 11 more sections

Key Result

Lemma 1

Let $\mathcal{C}$ be a $d$-deletion-correcting codebook satisfying equation eq:pairwise_lcs_constraint. The LCS-based decoder eq:lcs_decoder uniquely recovers the transmitted codeword from any received sequence $y \in \mathcal{Y}$.

Theorems & Definitions (3)

  • Remark 1
  • Lemma 1
  • proof